We review the mathematically rigorous formulation of the quantum theory of a linear field propagating in a globally hyperbolic spacetime. This formulation is accomplished via the algebraic approach, which, in essence, simultaneously admits all states in all possible (unitarily inequivalent) Hilbert space constructions. The physically nonsingular states are restricted by the requirement that their two-point function satisfy the Hadamard condition, which insures that the ultra-violet behavior of the state be similar to that of the vacuum state in Minkowski spacetime, and that the expected stress-energy tensor in the state be finite. We briefly review the Unruh and Hawking effects from the perspective of the theoretical framework adopted here. A brief discussion also is given of several open issues and questions in quantum field theory in curved spacetime regarding the treatment of ``back-reaction", the validity of some version of the ``averaged null energy condition'', and the formulation and properties of quantum field theory in causality violating spacetimes.
[1]
Hawking.
Quantum coherence and closed timelike curves.
,
1995,
Physical review. D, Particles and fields.
[2]
N. D. Birrell,et al.
Quantum fields in curved space
,
2007
.
[3]
Marek J. Radzikowski.
THE HADAMARD CONDITION AND KAY'S CONJECTURE IN (AXIOMATIC) QUANTUM FIELD THEORY ON CURVED SPACE-TIME
,
1992
.
[4]
R. Wald.
Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics
,
1994
.
[5]
R. Sorkin,et al.
A Positive mass theorem based on the focusing and retardation of null geodesics
,
1993,
gr-qc/9301015.